Quantum phase transitions in low-dimensional weakly interacting disordered Bose gases

In this thesis, we study the quantum phase transition triggered by an external random po- tential in ultra-cold low-dimensional weakly-interacting Bose gases at zero temperature. In one-dimensional systems, the quantum phases are characterized by the decay at long-range of the one-body density matrix. This decay exhibits an algebraic behaviour in the superfluid quasi-condensed phase while in the insulating Bose glass phase this it becomes exponen- tial, signaling the complete loss of coherence in the system. In two-dimensional systems, a characterization based on the long-range behaviour of the one-body density matrix appears to be highly demanding from a computational point of view. We therefore characterized the superfluid-insulator transition in two-dimensional systems by the low-energy behaviour of the cumulative density of states of the Bogoliubov excitations. While in the superfluid condensed phase this quantity grows as the square of the excitation energy in agreement with the existence of a finite velocity of sound, in the insulating Bose glass phase this power law is less than quadratic. This study is performed in the framework of an extended Bogoliubov approach properly adapted to treat low-dimensional Bose gases. Using a systematic numerical study, we draw the interaction-disorder phase diagrams of the superfluid-insulator transition in 1D and 2D. The phase boundary follows two different power- laws depending on the length scales characterizing the spatial correlations of the disorder and the strength of interactions. The power-law exponents were found to be in agreement with the ones predicted by scaling arguments, both in the white noise and the Thomas-Fermi regimes. In the two-dimensional system, the classical percolation threshold in the Thomas-Fermi limit was found to overestimate the critical disorder below which the onset of superfluidity should be observed for a given interaction strength.